Chapter 3 The K_0 Group of a Unital C* -algebra.

The Grothendieck construction 3.1.1
One can associate an abelian group to every abelian semigroup in a way analogous to how one obtains the integers from the naturals, and in much the same way as one obtains the rationals from the integers. We describe here how this works; the proofs of various statements are deferred to the next paragraph.
Let be an abelian semigroup. Define an equivalence relation on by if there exists such that
That is an equivalence relation is proved in paragraph 3.1.2.
Write for the quotient , and let denote the equivalence class in containing in .
The operation
is well-defined and turns into an abelian group. Note that and that for all .
The group is called the Grothendieck group of .
Let , the map is independent of the choice of , and is additive. It is called the Grothendieck map.

Properties of the Grothendieck group
The Grothendieck construction has the following properties.

1. Universal property
   If is an abelian group, and if is an additive map, then there is one, and only one group homomorphism making the diagram commute.
   
   
2. Functoriality
   To every additive map between semigroups and there is precisely one group homomorphism which makes the diagram commute.
   
   
4. Let be elements in . Then if and only if for some .
5. The Grothendieck map is injective if and only if has the cancellation property.
6. Let be an abelian group, and let be a non-empty subset of . If is closed under addition, then is an abelian semigroup with the cancellation property, is isomorphic to the subgroup generated by , and .

Example 3.1.3

1. The Grothendieck group of the abelian semigroup. is isomorphic to ; and has the cancellation property.
2. The Grothendieck group of the abelian semigroup is . And the semigroup has no cancellation.

Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Let be a Unital -algebra, and let be the abelian semigroup from Definition 2.3.3. (The semigroup of projections) Define to be the Grothendieck group of . i.e. Define by Where is the Grothendieck map.
So take any projection p, and send it to its K_0 class via the Grothendieck map on its equivalence class inside of the semigroup of projections .

The group K_00 3.1.5
The definition of given above also makes sense for non-unital -algebras, With the notation from Blackadar’s book. Let denote the Grothendieck group of the semigroup i.e. for every unital or non-unital - algebra .
For unital -algebras, the groups and are exactly the same, but in chapter 4 when we define the for a non-unital -algebra we will see that they disagree. One fatal flaw of the functor
is that it is not half exact.

Definition 3.1.6 (Stable equivalence)
Define a relation on as follows. If are projections in , then if and only if for some projection . The relation is called stable equivalence.

Characterization for stable equivalence
If is Unital, then we can characterize the stable equivalence condition nicely. if and only if for some positive integer
Proof:
If for some , then To prove this, just apply the properties given above for the sum of two elements in the semigroup of projections.

The standard picture of , described in the two propositions below, is a concrete and useful description of the -group of a unital -algebra. Proposition 3.1.8 shows that proposition 3.1.7 form a Universal property of .

Proposition 3.1.7 (The standard picture for K0 — the unital case) be a unital -algebra. Then Moreover,

1. for all projections Homomorphism
2. , where is the zero projection in . Zero preserving
3. If belong to for some and in , then Homotopic projections are K0 equivalent
4. If are mutually orthogonal projections in , then
5. For all in , if and only if K0 equivalence of projections is the same as stable equivalence of projections.
   Proof:
   The first identity follows from Properties of the Grothendieck group part 3.
   Hence if is an element in for a unital . we have that for some and some . Now choose . Letting and . Then are projections with and by Proposition 2.3.2. part 1. From this we deduce the second equality in the theorem. i.e. That .

Now we proof 1:
We know that Don’t forget here that gamma is the Grothendieck map that takes the abelian semigroup into the group completion.

Proof of 2:
We use the fact that from above we get that . This implies that .

Proof of 3:
Follows from the strength of homotopy over Murray von Neumann equivalence. Here the first two relations only hold when are living in the same matrix algebra over , and the other three relation are defined for all in The first implication holds by Proposition 2.2.7.

Proof of 4:
We have that by Proposition 2.3.2., and so by part 1.

Proof of 5:
If then there exists a projection such that by The Grothendieck construction 3.1.1.
Hence , so that .
Conversely, if then which in gives us that but since is a Group we can apply right-cancellation to get the desired result.

Proposition 3.1.8 (Universal property of K0)
Let be a unital -algebra , let be an abelian group, and suppose that is a map with the following properties:

1. for all projections
3. if belong to for some and in then
   THEN
   There exists a unique group homomorphism which makes the following diagram commute
   
   

The functor K0 for unital C-star Algebras 3.2.2The functor K00 3.2.3
Let and be unital -algebras, let be a star homomorphism . Associate to a group homomorphism as follows.
By Section 1.3, extends to a star homomorphism for each A star homomorphism maps projections to projections, and so
Define via for all , then satisfies 1,2,3 in Proposition 3.1.8 (Universal property of K0) and therefore factors uniquely through a group homomorphism given by For

If and are (not necessarily unital) algebras, then each star homomorphism induces a group homomorphism which satisfies for each To see this, first observe that Proposition 3.1.8 (Universal property of K0) can be extended word for word from to , then copy the argument above using the universal property.

This raises an important question, is this new functor K00, still not half- exact? we had noted that it was a brutal flaw of defining K0 for non-unital algebras. later in example 3.3.9 it will be proven that K00 is not half-exact…. but then in chapter 4 we get a new definition of K0 and K00 using the unitization, so should this paragraph above be ignored altogether?

Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)

1. For each unital -algebras, , .
2. If and are unital -algebras, and if and are -homomorphisms, then
3. .
4. For every pair of -algebras and , .
   Parts (1) and (2) say that is a functor from the category of unital -algebras to the category of Abelian groups. Parts (3) and (4) say that maps zero objects to zero objects, and zero morphisms to zero morphisms. WE INCLUDE THE ZERO C-ALGEBRA IN THE RANKS OF UNITAL C-ALGEBRAS!
   Proof:
   Use the definition of the functor
5. Take a map at the level of the algebras, , a * -homomorphism.
6. Induce a map on the matrix algebras of each one, .
7. Recall that it sends projections to projections and so it induces another map .
8. Lastly define a map from which after being put inside of the grothendieck map has the same properties as the universal property of as seen in Proposition 3.1.8 (Universal property of K0).
9. This forces it to factor through a unique map we denote as
   So now we check that .
   As desired.

Proposition 3.2.6 (Homotopy invariance of K0)
Let and be unital -algebras

1. If are homotopic -homomorphisms, then .
2. If and are homotopy equivalent, then is isomorphic to More specifically, if is a homotopy, then and are isomorphisms, and .

Proof:

1. Let be a point-wise continuous path of -homomorphisms connecting and . Extend this path to a pointwise continuous -homomorphisms for each .
   For every projection the path is continuous (pointwise continuous) and so . This shows that The map sends projections to projections, and these two projections are homotopic, so by shifting up to matrices, we get that they are K0 equivalent???
2. Follows from Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras) parts 1 and 2

Two -homomorphisms are said to be orthogonal to each other , or mutually orthogonal, if for all .
#mutually_orthogonal_star_hom

Lemma 3.2.7
If and are unital -algebras, and if are mutually orthogonal -homomorphisms, then is a -homomorphism and further, .

Lemma 3.2.8
For every unital -algebra the split exact sequence obtained by adjoining a unit to , induces a split exact sequence at the level of the groups. In other words, this is a split exact functor.

Paragraph 3.3.1 Traces and K0
Let be a -algebra, then a bounded trace on is a bounded linear map with the trace property: The trace property implies that if and are Murray-von Neumann equivalent, then .
A trace is positive if it maps positive elements to positive elements.
If a trace is positive and then is called a tracial state.

Any trace induces a natural trace on the matrix algebra by taking each
this trace on the matrix algebra then gives rise to a function and this function satisfies Proposition 3.1.8 (Universal property of K0) so there exists a unique group homomorphism which satisfies

#positive_trace
#tracial_state

Example 3.3.2
The group is isomorphic to for each positive integer . More specifically, if is the standard trace on then is an isomorphism. The cyclic group is generated by where is any one-dimensional projection in
This fact implies that the 0th k-theory of is since we can take .

Example 3.3.3
Let be an infinite dimensional hilbert space, then . The proof relies on the trace, for each we can find a projection onto a subspace of of dimension . Thus the semigroup we get is the integers union . The completion is .

Example 3.3.5
Let be an infinite dimensional hilbert space, then . The proof relies on the trace, for each we can find a projection onto a subspace of of dimension . Thus the semigroup we get is the integers union . The completion is .